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Year 2020, Volume: 3 Issue: 3, 117 - 127, 30.09.2020

Nutefe Kwami Agbeko

Abstract

References

  • [1] N.K. Agbeko, On optimal averages, Acta Math. Hungar. 63(2)(1994), 133–147.
  • [2] N.K. Agbeko, On the structure of optimal measures and some of its applications, Publ. Math. Debrecen 46(1-2)(1995), 79–87.
  • [3] V.I. Bogachev, Measure Theory, vol 1, Springer-Verlag Berlin Heidelberg 2007.
  • [4] C. Caratheodory, Vorlesungen über reelle Funktionen, Amer. Math. Soc. 2004.
  • [5] L. Drewnowski, A representation theorem for maxitive measures, Indag. Math. (N.S.), 20(1)(2009), 43–47.
  • [6] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Inc., 1992. ISBN: 0-8493-7157-0
  • [7] D. Harte, Multifractals. Theory and Applications, Chapman and Hall / Crc, Boca Raton London New York Washington D.C, 2001. ISBN 1-58488-154-2
  • [8] A.N. Kolmogorov, and S.V. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylack Press, Albany, N. Y. 1961.
  • [9] H. Lebesgue, Sur une generalistion de l’integrale definie, C.R. Acad.Sci. Paris 132(1901), 1025-1028.
  • [10] H. Lebesgue, Integrale, longueur, aire, Ann. Mat. Pura Appl. 7(1902), 231-359.
  • [11] J.C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City, 2011.
  • [12] N. Shilkret, Maxitive measure and integration, Indag. Math., 33(1971), 109-116.
  • [13] E.M. Taylor, Measure theory and integration, Graduate Studies in Mathematics, 76, 2006. ISBN: 13 978-0-8218-4180-8.

How to extend Carathéodory's theorem to lattice-valued functionals

Year 2020, Volume: 3 Issue: 3, 117 - 127, 30.09.2020

Nutefe Kwami Agbeko

Abstract

Substituting in the definition of outer measure the addition with the maximum (or the supremum, or the join) operation we obtain a new set function called retuo measure. It is proved that every retuo measure is an outer measure. We give necessary and sufficient conditions for a set function to be a retuo measure. Similarly as in the case of outer measure, we propose a way to construct retuo measures. We consider some theoretical applications for constructed pairs of outer and retuo measures in the image of the Hausdorff measure and dimension.

Keywords

outer mearsure sigma-algebra measurable sets Hausdorff measure Hausdorff dimension Caratheodory-type theorem

References

  • [1] N.K. Agbeko, On optimal averages, Acta Math. Hungar. 63(2)(1994), 133–147.
  • [2] N.K. Agbeko, On the structure of optimal measures and some of its applications, Publ. Math. Debrecen 46(1-2)(1995), 79–87.
  • [3] V.I. Bogachev, Measure Theory, vol 1, Springer-Verlag Berlin Heidelberg 2007.
  • [4] C. Caratheodory, Vorlesungen über reelle Funktionen, Amer. Math. Soc. 2004.
  • [5] L. Drewnowski, A representation theorem for maxitive measures, Indag. Math. (N.S.), 20(1)(2009), 43–47.
  • [6] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Inc., 1992. ISBN: 0-8493-7157-0
  • [7] D. Harte, Multifractals. Theory and Applications, Chapman and Hall / Crc, Boca Raton London New York Washington D.C, 2001. ISBN 1-58488-154-2
  • [8] A.N. Kolmogorov, and S.V. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylack Press, Albany, N. Y. 1961.
  • [9] H. Lebesgue, Sur une generalistion de l’integrale definie, C.R. Acad.Sci. Paris 132(1901), 1025-1028.
  • [10] H. Lebesgue, Integrale, longueur, aire, Ann. Mat. Pura Appl. 7(1902), 231-359.
  • [11] J.C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City, 2011.
  • [12] N. Shilkret, Maxitive measure and integration, Indag. Math., 33(1971), 109-116.
  • [13] E.M. Taylor, Measure theory and integration, Graduate Studies in Mathematics, 76, 2006. ISBN: 13 978-0-8218-4180-8.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Nutefe Kwami Agbeko

Publication Date September 30, 2020
Published in Issue Year 2020 Volume: 3 Issue: 3

Cite

APA Agbeko, N. K. (2020). How to extend Carathéodory’s theorem to lattice-valued functionals. Results in Nonlinear Analysis, 3(3), 117-127.
AMA Agbeko NK. How to extend Carathéodory’s theorem to lattice-valued functionals. RNA. September 2020;3(3):117-127.
Chicago Agbeko, Nutefe Kwami. “How to Extend Carathéodory’s Theorem to Lattice-Valued Functionals”. Results in Nonlinear Analysis 3, no. 3 (September 2020): 117-27.
EndNote Agbeko NK (September 1, 2020) How to extend Carathéodory’s theorem to lattice-valued functionals. Results in Nonlinear Analysis 3 3 117–127.
IEEE N. K. Agbeko, “How to extend Carathéodory’s theorem to lattice-valued functionals”, RNA, vol. 3, no. 3, pp. 117–127, 2020.
ISNAD Agbeko, Nutefe Kwami. “How to Extend Carathéodory’s Theorem to Lattice-Valued Functionals”. Results in Nonlinear Analysis 3/3 (September 2020), 117-127.
JAMA Agbeko NK. How to extend Carathéodory’s theorem to lattice-valued functionals. RNA. 2020;3:117–127.
MLA Agbeko, Nutefe Kwami. “How to Extend Carathéodory’s Theorem to Lattice-Valued Functionals”. Results in Nonlinear Analysis, vol. 3, no. 3, 2020, pp. 117-2.
Vancouver Agbeko NK. How to extend Carathéodory’s theorem to lattice-valued functionals. RNA. 2020;3(3):117-2.